<span class="mathjax-formula">$q$</span>-pseudoconvex and <span class="mathjax-formula">$q$</span>-complete domains

C ^OMPOSITIO M ^ATHEMATICA

G ^IUSEPPE V ^IGNA S ^URIA

q-pseudoconvex and q-complete domains

Compositio Mathematica, tome 53, n

^o

1 (1984), p. 105-111

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105

q-PSEUDOCONVEX

^and

q-COMPLETE

^DOMAINS

Giuseppe Vigna

^Suria

© 1984 Martinus Nijhoff Publishers, Dordrecht. Printed in The Netherlands.

Introduction

The Levi

problem

^was

originally posed

^{in the}

following

^{terms: if D is a}

domain in C " with

e2 boundary

^{which is}

pseudoconvex

^{is D a domain of}

holomorphy?

It was then realised that the

hypothesis

^{on the}

boundary

^{can be}

removed if

pseudoconvexity

^is

replaced by completeness,

^{which is a}

concept that makes sense in _any

analytic ^manifold,

and the final solution of the Levi

problem

^{due to Grauert} says that a

complete analytic

manifold is

necessarily

^Stein

[3].

The

original spirit

^{of the}

problem

^{has not been}

betrayed:

^domains

with

C2 boundary

^{in C" are}

pseudoconvex

^{if and}

only

^if

they

^are

complete ([4]

p.

50).

The same is not true any ^more if Cn is

replaced by

any

analytic

manifold: ^a well known

example

of Grauert

provides

^a subset with

C2 boundary

^{of a}

complex

^{torus which is}

pseudoconvex

^{but all}

holomorphic

functions thereon are constant.

In this _paper we prove that a

q-pseudoconvex

open subset of a Stein manifold is

necessarily q-complete (the

^converse is also true, ^see

[2]).

^This

seems to be one of those facts that _every

complex analyst believes, perhaps

^for

psycological

^{reasons, but no}

precise

^reference

is,

^to my

knowledge,

^{available and all} mathematicians whom 1 have asked so far don’t seem to know how a

precise proof

^should go; the modest aim of this _{paper is} to fill this _{gap and}

provide

^a definite reference.

Most of the ideas in the

proof

^{are due to Mike Eastwood to whom 1}

am, ^once more,

deeply grateful.

We

briefly

recall the basic definitions:

DEFINITION 1: Let D be an open subset of an

analytic

^{manifold M of}

dimension n ; ^we say that D has

C2 boundary

^if for all x E aD there exists

an open

neighbourhood

^{U of x and a}

^C2

^function (p:

U ~ R,

^called

defining function

^{of D at x s.t. D ~ U =}

(y

^{E U s.t.}

cp(y) 01 -

^and

106

d~(x) ~ 0;

ⁱⁿ these conditions we can consider the

complex

^Hessian

where zl, z2, ... , zn ^are local

holomorphic

coordinates at x. The

signature

of this Hermitian matrix does not

depend

^on the choice of the local

holomorphic

coordinates but it does

depend

^on cp. However the Levi

form

where

TxaD

⁼

{v

⁼

03A3m1v1~/~zl ~ Tx M

^{s. t.}

03A3n1vi~~/~zi(x)

⁼

0}

is the holo-

morphic

tangent space of aD at _x, has a

signature

^that

depends only

^on

D and x.

If

n ( x )

denotes the number of

negative eigenvalues of F(~)(x)

^we say that D is

q-pseudoconvex

^if

n(x)

q for all x E aD.

DEFINITION 2: A

complex

n-dimensional manifold D is said to be

q-complete

^{if we can find a}

q-plurisubharmonic

^exaustion

function

^{on D i.e.}

a

C2

function 03A8 : D ~ R s.t.

(1)

^{for all c ~ R the set}

Bc = {x ~ D

^s.t.

03A8(x) c}

^is

relatively compact in

^{D and}

(2)

^The

complex

^Hessian

H(03A8)(x)

^{has at}

least n - q positive eigen-

values for all x in D.

0-pseudoconvex

^and

0-complete

^{domains are}

simply

^called

pseudocon-

vex and

complete.

THEOREM:

If D is a

domain with

C2 boundary

^{in a Stein}

manifold

^{M and D}

is

q-pseudoconvex

^{then it is also}

q-complete.

PROOF: We shall divide the

proof

into several steps.

Step

^{1: As there is}

always

^an

analytic embedding

^{of M into C}

N,

^{for some}

large

^N

(see [5]

p.

359)

^{we can} suppose ^{at once} that M is an

analytic

submanifold

of CN.

Choose a

holomorphic

^tubular

neighbourhood

p : V

~ M and set

D = p-1(D) (cfr. [1] proof

^{of Lemma}

1,

p.

131).

^{We claim}

that,

^after

shrinking

if necessary,

(a)

^{~x ~}

ah _{n V,} aD

is

C2

at _x,

(b)

^{If we consider}

D

as an open subset

of CN

then

n ( x, ) =

n(p(x), D),

^{for al x E}

~

n V.

Indeed,

^{since the}

problem

^{is local we can} suppose that local coordi- nates z1, z2, ...,zN have been chosen s.t., ^near x, M ⁼

( z

^S.t. zN-n+1 ⁼ ZN-n+2 ^=... = zN

= 0),

Zl’ Z2"" Zn ^are local coordinates of M at x and

p Z1, z2, ... , ZN ) = (z1,

z2, ... , zn,

0,...,0).

Let

Û

be a

neighbourhood

^{of x}

^{in CN}

^so small that zl, z2,,..., zN ^are defined in

Û

and that there exists ^a

C2 defining

function (D : U =

~

M - R for D with

d03A6(x) ~

^{0 and}

U c

V.

By shrinking Û

^if necessary ^we

can also _suppose that

~ p-1(U).

Define : ~ R by = 03A6 o p

^i.e.

(z1,

z2, ... ,

zN) = 03A6(z1,

z2, ... , 1 z,,,

0, ... , ,0). Then

is a

defining

function for

b

at x.

Moreover

where as usual v ⁼

03A3Ni=1vi~/~zi,

^and

This _proves the claim.

Step

^2:

^So,

^{if we} suppose that D is

q-pseudoconvex

^{we have}

that,

Vx OE V ~

aD, ^ab

^is

^e2

^{at x and}

n(x, )

q.

Consider the function _{p :}

CN ~ R given by

Where dist denotes the Euclidean distance. Since Vx E=-

aD, ^aD

^is

^C2

at x, we can conclude that there exists ^a

neighbourhood Û’

^{of aD}

^{in CN}

^on

which p is

C2 (by

the inverse function

theorem).

By shrinking Û’

if necessary, ^{we can} also _suppose that

Vy

ⁱⁿ

U’

there exists

exactly

^one

point c(y) ~ aD n U’

which is the closest

point

^{to y}

under the Euclidean

distance,

^that

d03C1(c(y))

^~ 0 and that

n(c(y), )

q.

Let (P:

Û’ ~ ~ R

be the function _(p ⁼

log

^{p; we} claim that the Hessian

(£cp)(y)

^{has at most} q

positive eigenvalues B;/y.

Indeed _suppose that this is

false,

i.e. there exists a

point y

ⁱⁿ

U’

ⁿ

D

s.t.

(H~)(y)

^has

(at least) q + 1 positive eigenvalues;

^the

geometric interpretation

is: there are linear coordinates

(t1, t2l 1 N) ^{of CN}

^{s.t. the}

Hermitian form

given by

^{the matrix}

108

is

positive

^{definite on} the linear

subspace V

^of

TvCN = CN spanned by

(~/~1, ~/~t2,...,~/~tq+1).

By Taylor’s

^{theorem we have}

where a, = 1 2~~/~ti(y)

^and

bjk

⁼

~2~(y)/~tj~tk

^are constants, and

0(|t|2)

has the

property

^that

lim t ~00(|t|2)/|t|2

^{= 0 and so also}

In order to

simplify

notation omit the limits of the summands and write

A(t)

= y ⁺

03A3tj~/~tj , ^B(t)

⁼

exp(03A3aiti

⁺

Ebjktjtk)’

Then the above

equality

^can be written as

03C1(A(t)) - 03C1(y)(B(t)| =

{exp03A3Cjktjtk

⁺

0(|t|2)) -1} 03C1(y)|B(t)| = {03A3Cjktjtk

⁺

0’(|t|2)}03C1(y)|B(t)|,

where the last

equality

^{is obtained}

by expanding

ⁱⁿ

Taylor

^{series the}

function _exp and

0’(|t|2)

^{has the same}

properties

^as

0(|t|2).

^{Then one has}

so we can choose E &#x3E; 0 small

enough

^{s. t.}

~t, |t|

^{~, one has}

(a) A(t) ~ ~ ’

^and

(b) p (A (t» - 03C1(y)|B(t) &#x3E; 03C1 (y)/2 · 03A3Cjktjtk.

Set u =

c(y)-y

and define an

analytic

^{function T on the} open ball

B~ = {t ~ Cq+1 ^S.t. |t| ~}:

We can also _suppose that E is so small that

T( t )

^E

U’

if t ~

B~.

^{Then it}

is easy ^to

check,

^{and a}

picture

^shows

how,

^{that if t}

e B,

^{one has}

(c) 03C1(T’(t)) 03C1(A(t)) - |u||B(t)| |u|/203A3Cjktjtk 0

This in

particular

proves that

T( t ) E D

^{for all t E}

B~ - {0}, ^and,

^since

03C1(T(0)) = 03C1(c(y0) = 0,

^{0 is a minimum for} the function _po

T : B~ ~ R,

and _so,

taking partial derivatives,

Using

^{the chain rule and the} fact that T is

analytic

^{we have:}

(d) 03A3Nh=1~03C1/~zh(c(y))~Th/~tj(0) = 0 for j = 1,2, .., q + 1.

In other words the vectors

~T/~tj (0), j

⁼

1, 2,...,

q ⁺ 1, ^{are in}

Tc( v)~.

Moreover,

^B;/t

in Cq+1,

^{we have}

(e) 03A3q+1j,k=1~203C1o T(0)/~tj~tktjtk |u|/403A3q+1j,k=1Cjktjtk.

To prove this we first observe that it is

clearly enough

^to check it for

small 1 t 1.

From the above

inequality (c), using Taylor series,

^{we deduce}

for all

t ~ B~,

^where

djk

⁼

^~203C1o T(0)/~tj~tk

^{are constants and}

0"(ltI2)

^has

the ^same

properties

^as

0(|t|2).

Then,

after

reducing E

^if necessary, ^we

have,

^{Vt E=-}

B,,

Let

/J

⁼

ei03B8tj

^for

0 03B8 2 03C0; writing t’

ⁱⁿ the above

inequality

^and

observing

that the second and third term are

unchanged

^{under the}

substitution t ^-

t’,

^we

deduce, V0,

and

by choosing B

^{so that} the first term is

negative

^we prove the

inequality (e).

Using again

the chain rule and the fact that T is

analytic

^{we have that}

the Hermitian form

is

positive

^definite.

It follows

easily

that the Hermitian form

(~203C1(c(y))/~zh~zm)Nh,m=1

^is

positive

^{definite on} the linear

subspace

^{V of}

7c(v)~ spanned by

^the

110

vectors

~T/~Tj(0), j = 1, 2,..., q + 1 ;

ⁱⁿ

particular

it follows automati-

cally

that these vectors are

linearly independent,

^{so that}

^{dimV = q}

⁺

^1;

but

since - p is a defining

function for D at

c(y),

^{we have that}

n(c(y), ) q + 1

^and this contradicts our

hypothesis,

^so that the claim is

proved.

Step

^3:

By restricting ~

^to

Û’

n D we find a

C2 function,

called

again

~:W = ’ ~ D ~ R

^s.t.

(a) limy~~D~(y) = - ~,

(b) (,)glip)(y)

^{has at} most q

positive eigenvalues Vy

^{in W.}

Let F be a closed subset of M s.t. D - W ~ int F c F c

D,

and let

0 03A8 1 be a C2 bump

^{function s.t. ’1’ = 0}

on F, ’1’

^{= 1 in a}

neighbour-

hood of M -

D,

^and supppse that F is chosen so

that ~(y)

⁰

for y 5É

^F.

By considering

the function

qq’

= ~ ’

P,

^{we have that}

(a) limy ~~D~’(y) = - ~,

(b) (H~’)(y)

^{has at}

most q positive eigenvalues ~y ~

^{D -}

F, (c) w’

^0.

Now we use the fact that M is Stein and ^so

0-complete (see [5],

^lemma

p.

358)

i.e. there exists a

0-plurisubharmonic

exhaustion function À : M - R .

’fin E

Z,

the set

Kn

⁼

{y ~

^{M s.t.}

03BB(y) n}

^{is compact, therefore so is}

F n

Kn

and there exist constants

Cn

^s.t.

Now choose a

C2

function

f :

^{R - R with the}

properties

First we notice

that, Vc E=- R, Be = {y ~

^{D s. t.}

x(y) cl

^is

^contained, by

^the property

( c)

^of

qq’

ⁱⁿ

f y

^{E D s.t.}

f o 03BB (y) c}

^{which is}

^compact by

^the

assumptions

^on

f

^{and À.}

Moreover Bc

^{is closed in D}

^and,

^since

limy~~D’(y)

^{= - oo,} it is also closed in M.

Thus Bc

^is

compact and X

^is

an exhaustion function.

For

all y E

^{D we have}

where A(y) = (~03BB/~zi(y) · ~03BB/~zj(y))ni,j=1 ^is

^a

semipositive

^Hermitian

form. If

y E D - F

then there exists a linear

subspace

^of

7§,D,

^of

dimension _{n - q}

where -(H~~’)(y)

^is

positive

semidefinite.

Therefore

(H~)(y)

^is

positive

^{definite on V.}

If y E

^{F then}

either y E Ko

^~ F in which case

or y E

(Kn+1 - Kn )

^{~ F for some}

integer n ^0,

^{in which case}

Therefore X

^{is also}

q-plurisubharmonic

^{and we can}

finally

say that the

theorem is

proved.

⁰

References

[1] K. DIEDERICH and J.E. FORNAESS: Pseudoconvex domains: bounded strictly plurisub-

harmonic exhaustion functions. Inv. Math. 39 (1977) ^129-141.

[2] M.G. EASTWOOD and G. VIGNA SURIA: Cohomologically complete ^and pseudoconvex

domains. Comm. Math. Helv. 55 (1980) ^413-426.

[3] H. GRAUERT: On Levi’s problem ^{and the} embedding ^{of real} analytic manifolds. Ann.

Math. 68 (1958) ^460-472.

[4] L. HORMANDER: An introduction to complex analysis ⁱⁿ several variables. Princeton N.J.

Van Nostrand (1966).

[5] R. NARASIMHAN: The Levi problem ^for complex spaces. Math. Ann. 142 (1961) 355-365.

(Oblatum 2-XI-1982) Giuseppe Vigna ^Suria Dip. di Matematica Fac. di Scienze Università di Trento 38050 Povo (TN) Italy